Exact solutions for analog Hawking effect in dielectric media

TL;DR

This paper derives exact solutions for the analog Hawking effect in dielectric media using Fuchsian differential equations, providing insights into thermality and Hawking temperature without relying on weak dispersion approximations.

Contribution

It introduces a method to obtain exact solutions for the analog Hawking effect in dielectric media, involving hypergeometric functions and comprehensive connection formulas, without weak dispersion assumptions.

Findings

Exact solutions involve generalized hypergeometric functions.

Hawking temperature matches previous approximate results.

Full analysis of the Stokes phenomenon and connection formulas.

Abstract

In the framework of the analog Hawking radiation for dielectric media, we analyze a toy-model and also the 2D reduction of the Hopfield model for a specific monotone and realistic profile for the refractive index. We are able to provide exact solutions, which do not require any weak dispersion approximation. The theory of Fuchsian ordinary differential equations is the basic tool for recovering exact solutions, which are rigoroulsy identified, and involve the so-called generalized hypergeometric functions $_4F_3(\alpha_1,\alpha_2,\alpha_3,\alpha_4;\beta_1,\beta_2,\beta_3;z)$. A complete set of connection formulas are available, both for the subcritical case and for the transcritical one, and also the Stokes phenomenon occurring in the problem is fully discussed. From the physical point of view, we focus on the problem of thermality. Under suitable conditions, the Hawking temperature is deduced, and we show that it is in fully agreement with the expression deduced in other frameworks under various approximations.